Initial velocity excitations are straightforward in the DW paradigm,
but can be less intuitive in the FDTD domain. It is well known that
velocity in a displacement-wave DW simulation is determined by the
difference of the right- and left-going waves
[25]. Specifically, initial velocity waves can
be computed from from initial displacement waves by spatially
differentiating to obtain traveling slope waves, multiplying by minus the tension to obtain force
waves, and finally dividing by the wave impedance to
obtain velocity waves:

(14)

where
denotes sound speed. The initial string
velocity at each point is then
. (A
more direct derivation can be based on differentiating Eq. (5)
with respect to and solving for velocity traveling-wave
components, considering left- and right-going cases separately at
first, and arguing the general case by superposition.)

We can see from Eq. (12) that such asymmetry can be caused by
unequal weighting of and
. For example, the
initialization

corresponds to an impulsevelocity excitation at position
. In this case, both interleaved grids are excited.